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cos(x-sinx)sqrt(3x-x^2)>=0 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src]
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cos(x - sin(x))*\/  3*x - x   >= 0
$$\sqrt{- x^{2} + 3 x} \cos{\left(x - \sin{\left(x \right)} \right)} \geq 0$$
sqrt(-x^2 + 3*x)*cos(x - sin(x)) >= 0
Solución detallada
Se da la desigualdad:
$$\sqrt{- x^{2} + 3 x} \cos{\left(x - \sin{\left(x \right)} \right)} \geq 0$$
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
$$\sqrt{- x^{2} + 3 x} \cos{\left(x - \sin{\left(x \right)} \right)} = 0$$
Resolvemos:
$$x_{1} = -2.30988146001006$$
$$x_{2} = -79.3715275333246$$
$$x_{3} = 35.3892303830675$$
$$x_{4} = -21.1594373815488$$
$$x_{5} = 71.4249198389855$$
$$x_{6} = -91.9378981476837$$
$$x_{7} = 0.916089372113664 + 1.8093613412957 i$$
$$x_{8} = 14.8762520743692$$
$$x_{9} = 2.30988146001006$$
$$x_{10} = -16.5396744615287$$
$$x_{11} = -85.6547128405042$$
$$x_{12} = -54.2387863046062$$
$$x_{13} = 96.5576610677038$$
$$x_{14} = 41.672415690247$$
$$x_{15} = 33.725807995908$$
$$x_{16} = -41.672415690247$$
$$x_{17} = -98.2210834548633$$
$$x_{18} = -83.9912904533447$$
$$x_{19} = -46.2921786102672$$
$$x_{20} = 98.2210834548633$$
$$x_{21} = -3.97330384716953$$
$$x_{22} = -66.8051569189654$$
$$x_{23} = 40.0089933030876$$
$$x_{24} = 3.97330384716953$$
$$x_{25} = 54.2387863046062$$
$$x_{26} = -10.2564891543491$$
$$x_{27} = -40.0089933030876$$
$$x_{28} = -96.5576610677038$$
$$x_{29} = 27.4426226887284$$
$$x_{30} = 47.9556009974266$$
$$x_{31} = -29.1060450758879$$
$$x_{32} = 10.2564891543491$$
$$x_{33} = 83.9912904533447$$
$$x_{34} = 91.9378981476837$$
$$x_{35} = -27.4426226887284$$
$$x_{36} = 0.916089372113664 - 1.8093613412957 i$$
$$x_{37} = 52.5753639174468$$
$$x_{38} = 79.3715275333246$$
$$x_{39} = 66.8051569189654$$
$$x_{40} = 85.6547128405042$$
$$x_{41} = 0$$
$$x_{42} = -8.59306676718964$$
$$x_{43} = 16.5396744615287$$
$$x_{44} = -47.9556009974266$$
$$x_{45} = 22.8228597687083$$
$$x_{46} = -0.916089372113664 - 1.8093613412957 i$$
$$x_{47} = -33.725807995908$$
$$x_{48} = -90.2744757605243$$
$$x_{49} = 90.2744757605243$$
$$x_{50} = -52.5753639174468$$
$$x_{51} = -60.5219716117858$$
$$x_{52} = -77.7081051461651$$
$$x_{53} = -73.088342226145$$
$$x_{54} = -35.3892303830675$$
$$x_{55} = 46.2921786102672$$
$$x_{56} = 8.59306676718964$$
$$x_{57} = 77.7081051461651$$
$$x_{58} = -71.4249198389855$$
$$x_{59} = 58.8585492246263$$
$$x_{60} = 60.5219716117858$$
Descartamos las soluciones complejas:
$$x_{1} = -2.30988146001006$$
$$x_{2} = -79.3715275333246$$
$$x_{3} = 35.3892303830675$$
$$x_{4} = -21.1594373815488$$
$$x_{5} = 71.4249198389855$$
$$x_{6} = -91.9378981476837$$
$$x_{7} = 14.8762520743692$$
$$x_{8} = 2.30988146001006$$
$$x_{9} = -16.5396744615287$$
$$x_{10} = -85.6547128405042$$
$$x_{11} = -54.2387863046062$$
$$x_{12} = 96.5576610677038$$
$$x_{13} = 41.672415690247$$
$$x_{14} = 33.725807995908$$
$$x_{15} = -41.672415690247$$
$$x_{16} = -98.2210834548633$$
$$x_{17} = -83.9912904533447$$
$$x_{18} = -46.2921786102672$$
$$x_{19} = 98.2210834548633$$
$$x_{20} = -3.97330384716953$$
$$x_{21} = -66.8051569189654$$
$$x_{22} = 40.0089933030876$$
$$x_{23} = 3.97330384716953$$
$$x_{24} = 54.2387863046062$$
$$x_{25} = -10.2564891543491$$
$$x_{26} = -40.0089933030876$$
$$x_{27} = -96.5576610677038$$
$$x_{28} = 27.4426226887284$$
$$x_{29} = 47.9556009974266$$
$$x_{30} = -29.1060450758879$$
$$x_{31} = 10.2564891543491$$
$$x_{32} = 83.9912904533447$$
$$x_{33} = 91.9378981476837$$
$$x_{34} = -27.4426226887284$$
$$x_{35} = 52.5753639174468$$
$$x_{36} = 79.3715275333246$$
$$x_{37} = 66.8051569189654$$
$$x_{38} = 85.6547128405042$$
$$x_{39} = 0$$
$$x_{40} = -8.59306676718964$$
$$x_{41} = 16.5396744615287$$
$$x_{42} = -47.9556009974266$$
$$x_{43} = 22.8228597687083$$
$$x_{44} = -33.725807995908$$
$$x_{45} = -90.2744757605243$$
$$x_{46} = 90.2744757605243$$
$$x_{47} = -52.5753639174468$$
$$x_{48} = -60.5219716117858$$
$$x_{49} = -77.7081051461651$$
$$x_{50} = -73.088342226145$$
$$x_{51} = -35.3892303830675$$
$$x_{52} = 46.2921786102672$$
$$x_{53} = 8.59306676718964$$
$$x_{54} = 77.7081051461651$$
$$x_{55} = -71.4249198389855$$
$$x_{56} = 58.8585492246263$$
$$x_{57} = 60.5219716117858$$
Las raíces dadas
$$x_{16} = -98.2210834548633$$
$$x_{27} = -96.5576610677038$$
$$x_{6} = -91.9378981476837$$
$$x_{45} = -90.2744757605243$$
$$x_{10} = -85.6547128405042$$
$$x_{17} = -83.9912904533447$$
$$x_{2} = -79.3715275333246$$
$$x_{49} = -77.7081051461651$$
$$x_{50} = -73.088342226145$$
$$x_{55} = -71.4249198389855$$
$$x_{21} = -66.8051569189654$$
$$x_{48} = -60.5219716117858$$
$$x_{11} = -54.2387863046062$$
$$x_{47} = -52.5753639174468$$
$$x_{42} = -47.9556009974266$$
$$x_{18} = -46.2921786102672$$
$$x_{15} = -41.672415690247$$
$$x_{26} = -40.0089933030876$$
$$x_{51} = -35.3892303830675$$
$$x_{44} = -33.725807995908$$
$$x_{30} = -29.1060450758879$$
$$x_{34} = -27.4426226887284$$
$$x_{4} = -21.1594373815488$$
$$x_{9} = -16.5396744615287$$
$$x_{25} = -10.2564891543491$$
$$x_{40} = -8.59306676718964$$
$$x_{20} = -3.97330384716953$$
$$x_{1} = -2.30988146001006$$
$$x_{39} = 0$$
$$x_{8} = 2.30988146001006$$
$$x_{23} = 3.97330384716953$$
$$x_{53} = 8.59306676718964$$
$$x_{31} = 10.2564891543491$$
$$x_{7} = 14.8762520743692$$
$$x_{41} = 16.5396744615287$$
$$x_{43} = 22.8228597687083$$
$$x_{28} = 27.4426226887284$$
$$x_{14} = 33.725807995908$$
$$x_{3} = 35.3892303830675$$
$$x_{22} = 40.0089933030876$$
$$x_{13} = 41.672415690247$$
$$x_{52} = 46.2921786102672$$
$$x_{29} = 47.9556009974266$$
$$x_{35} = 52.5753639174468$$
$$x_{24} = 54.2387863046062$$
$$x_{56} = 58.8585492246263$$
$$x_{57} = 60.5219716117858$$
$$x_{37} = 66.8051569189654$$
$$x_{5} = 71.4249198389855$$
$$x_{54} = 77.7081051461651$$
$$x_{36} = 79.3715275333246$$
$$x_{32} = 83.9912904533447$$
$$x_{38} = 85.6547128405042$$
$$x_{46} = 90.2744757605243$$
$$x_{33} = 91.9378981476837$$
$$x_{12} = 96.5576610677038$$
$$x_{19} = 98.2210834548633$$
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
$$x_{0} \leq x_{16}$$
Consideremos, por ejemplo, el punto
$$x_{0} = x_{16} - \frac{1}{10}$$
=
$$-98.2210834548633 + - \frac{1}{10}$$
=
$$-98.3210834548633$$
lo sustituimos en la expresión
$$\sqrt{- x^{2} + 3 x} \cos{\left(x - \sin{\left(x \right)} \right)} \geq 0$$
$$\sqrt{- \left(-98.3210834548633\right)^{2} + \left(-98.3210834548633\right) 3} \cos{\left(-98.3210834548633 - \sin{\left(-98.3210834548633 \right)} \right)} \geq 0$$
16.2518727724411*I >= 0

Entonces
$$x \leq -98.2210834548633$$
no se cumple
significa que una de las soluciones de nuestra ecuación será con:
$$x \geq -98.2210834548633 \wedge x \leq -96.5576610677038$$
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       x16      x27      x6      x45      x10      x17      x2      x49      x50      x55      x21      x48      x11      x47      x42      x18      x15      x26      x51      x44      x30      x34      x4      x9      x25      x40      x20      x1      x39      x8      x23      x53      x31      x7      x41      x43      x28      x14      x3      x22      x13      x52      x29      x35      x24      x56      x57      x37      x5      x54      x36      x32      x38      x46      x33      x12      x19

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
$$x \geq -98.2210834548633 \wedge x \leq -96.5576610677038$$
$$x \geq -91.9378981476837 \wedge x \leq -90.2744757605243$$
$$x \geq -85.6547128405042 \wedge x \leq -83.9912904533447$$
$$x \geq -79.3715275333246 \wedge x \leq -77.7081051461651$$
$$x \geq -73.088342226145 \wedge x \leq -71.4249198389855$$
$$x \geq -66.8051569189654 \wedge x \leq -60.5219716117858$$
$$x \geq -54.2387863046062 \wedge x \leq -52.5753639174468$$
$$x \geq -47.9556009974266 \wedge x \leq -46.2921786102672$$
$$x \geq -41.672415690247 \wedge x \leq -40.0089933030876$$
$$x \geq -35.3892303830675 \wedge x \leq -33.725807995908$$
$$x \geq -29.1060450758879 \wedge x \leq -27.4426226887284$$
$$x \geq -21.1594373815488 \wedge x \leq -16.5396744615287$$
$$x \geq -10.2564891543491 \wedge x \leq -8.59306676718964$$
$$x \geq -3.97330384716953 \wedge x \leq -2.30988146001006$$
$$x \geq 0 \wedge x \leq 2.30988146001006$$
$$x \geq 3.97330384716953 \wedge x \leq 8.59306676718964$$
$$x \geq 10.2564891543491 \wedge x \leq 14.8762520743692$$
$$x \geq 16.5396744615287 \wedge x \leq 22.8228597687083$$
$$x \geq 27.4426226887284 \wedge x \leq 33.725807995908$$
$$x \geq 35.3892303830675 \wedge x \leq 40.0089933030876$$
$$x \geq 41.672415690247 \wedge x \leq 46.2921786102672$$
$$x \geq 47.9556009974266 \wedge x \leq 52.5753639174468$$
$$x \geq 54.2387863046062 \wedge x \leq 58.8585492246263$$
$$x \geq 60.5219716117858 \wedge x \leq 66.8051569189654$$
$$x \geq 71.4249198389855 \wedge x \leq 77.7081051461651$$
$$x \geq 79.3715275333246 \wedge x \leq 83.9912904533447$$
$$x \geq 85.6547128405042 \wedge x \leq 90.2744757605243$$
$$x \geq 91.9378981476837 \wedge x \leq 96.5576610677038$$
$$x \geq 98.2210834548633$$
Solución de la desigualdad en el gráfico